Yin Tian

Abstract

We define the Hopf superalgebra
UT(sl(1|1)),
which is a variant of the quantum supergroup
Uq(sl(1|1)), and its
representations
V1⊗n for
n>0. We construct families
of DG algebras
A,
B and
Rn, and consider the
DG categories
DGP(A),
DGP(B) and
DGP(Rn),
which are full DG subcategories of the categories of DG
A–,
B– and
Rn–modules
generated by certain distinguished projective modules. Their
0 th homology
categories
HP(A),
HP(B) and
HP(Rn) are triangulated
and give algebraic formulations of the contact categories of an annulus, a twice punctured
disk and an n
times punctured disk. Their Grothendieck groups are isomorphic to
UT(sl(1|1)),
UT(sl(1|1))⊗ℤUT(sl(1|1)) and
V1⊗n,
respectively. We categorify the multiplication and comultiplication on
UT(sl(1|1)) to a bifunctor
HP(A)× HP(A)→ HP(A) and a functor
HP(A)→ HP(B), respectively.
The
UT(sl(1|1))–action
on
V1⊗n is lifted to
a bifunctor
HP(A)× HP(Rn)→ HP(Rn).